Continuity of Parametric Optima for Possibly Discontinuous Functions and Noncompact Decision Sets

TL;DR

This paper extends Berge's maximum theorem to cases with discontinuous objective functions and noncompact feasible sets, providing necessary and sufficient conditions for the continuity of value functions and solutions in complex optimization problems.

Contribution

It generalizes Berge's maximum theorem by relaxing continuity and compactness assumptions, offering broader applicability in operations research and economics.

Findings

Provides necessary and sufficient conditions for continuity.

Applies results to inventory control problems.

Analyzes robust optimization with noncompact sets.

Abstract

This paper investigates continuity properties of value functions and solutions for parametric optimization problems. These problems are important in operations research, control, and economics because optimality equations are their particular cases. The classic fact, Berge's maximum theorem, gives sufficient conditions for continuity of value functions and upper semicontinuity of solution multifunctions. Berge's maximum theorem assumes that the objective function is continuous and the multifunction of feasible sets is compact-valued. These assumptions are not satisfied in many applied problems, which historically has limited the relevance of the theorem. This paper generalizes Berge's maximum theorem in three directions: (i) the objective function may not be continuous, (ii) the multifunction of feasible sets may not be compact-valued, and (iii) necessary and sufficient conditions are provided. To illustrate the main theorem, this paper provides applications to inventory control and to the analysis of robust optimization over possibly noncompact action sets and discontinuous objective functions.